# Borcherds lifts and Maaß lifts on the paramodular group of level 3

• Borcherds Lifts und Maaß Lifts zur paramodularen Gruppe der Stufe 3

Kreuzer, Judith; Krieg, Aloys (Thesis advisor)

Aachen (2014)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2014

Abstract

The main goal of this thesis is to give a full characterization of those modular forms on the paramodular group Gamma_3^+ of level 3 that are Maaß lifts and Borcherds lifts at the same time. The Maaß lift is an additive lift that yields a Fourier expansion. Moreover, we obtain a certain relation among the Fourier coefficients. In contrast, the Borcherds lift yields an infinite product expansion of the modular form. Thus, it is multiplicative and we can determine the divisors of a Borcherds lift quite easily. By construction, we already know certain properties of a modular form obtained by a Maaß lift or a Borcherds lift. But these properties differ a lot in structure, which makes it difficult to compare the two lifts. The question if a modular form is a Maaß lift and a Borcherds lift at the same time has hardly been examined until now. A first systematical approach to this question was taken by B. Heim and A. Murase in the case of Siegel modular forms of degree 2 without character. In answering the question of simultaneous lifts for the paramodular group Gamma_3^+ of level 3, it turned out that it is not possible to give an analogous proof to the Siegel case. Thus, a new proof was developed. This new approach is presented first by characterizing simultaneous lifts in the case of Siegel modular forms with characters. Afterwards, the same line of argument will be used for the paramodular case of level 3. An important step in describing simultaneous lifts lies in finding properties of Maaß lifts and Borcherds lifts on Gamma_3^+. Some of these characteristics turn out to be interesting in their own right. For example, a Maaß lift on Gamma_3^+ can have only certain characters, and a method to construct Borcherds lifts with any given divisor is presented. Moreover, it is possible to give structural statements on these constructed Borcherds lifts without knowing them in concrete terms; e.g. if it is a cusp form or if it has trivial character. In addition to that, one can show that every non-cuspidal Borcherds lift on Gamma_3^+ must have trivial character. Finally, it turns out that -up to a multiplicative constant- there are exactly six modular forms on Gamma_3^+ that are Maaß lifts and Borcherds lifts at the same time.